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The conjectural theory of mixed motives would be a universal
cohomology theory in arithmetic algebraic geometry. The monograph
describes the approach to motives via their well-defined
realizations. This includes a review of several known cohomology
theories. A new absolute cohomology is introduced and
studied.
The book assumes knowledge of the standard cohomological techniques
in algebraic geometry as well as K-theory. So the monograph is
primarily intended for researchers. Advanced graduate students can
use it as a guide to the literature.
This book casts the theory of periods of algebraic varieties in the
natural setting of Madhav Nori's abelian category of mixed motives.
It develops Nori's approach to mixed motives from scratch, thereby
filling an important gap in the literature, and then explains the
connection of mixed motives to periods, including a detailed
account of the theory of period numbers in the sense of
Kontsevich-Zagier and their structural properties. Period numbers
are central to number theory and algebraic geometry, and also play
an important role in other fields such as mathematical physics.
There are long-standing conjectures about their transcendence
properties, best understood in the language of cohomology of
algebraic varieties or, more generally, motives. Readers of this
book will discover that Nori's unconditional construction of an
abelian category of motives (over fields embeddable into the
complex numbers) is particularly well suited for this purpose.
Notably, Kontsevich's formal period algebra represents a torsor
under the motivic Galois group in Nori's sense, and the period
conjecture of Kontsevich and Zagier can be recast in this setting.
Periods and Nori Motives is highly informative and will appeal to
graduate students interested in algebraic geometry and number
theory as well as researchers working in related fields. Containing
relevant background material on topics such as singular cohomology,
algebraic de Rham cohomology, diagram categories and rigid tensor
categories, as well as many interesting examples, the overall
presentation of this book is self-contained.
This book casts the theory of periods of algebraic varieties in the
natural setting of Madhav Nori's abelian category of mixed motives.
It develops Nori's approach to mixed motives from scratch, thereby
filling an important gap in the literature, and then explains the
connection of mixed motives to periods, including a detailed
account of the theory of period numbers in the sense of
Kontsevich-Zagier and their structural properties. Period numbers
are central to number theory and algebraic geometry, and also play
an important role in other fields such as mathematical physics.
There are long-standing conjectures about their transcendence
properties, best understood in the language of cohomology of
algebraic varieties or, more generally, motives. Readers of this
book will discover that Nori's unconditional construction of an
abelian category of motives (over fields embeddable into the
complex numbers) is particularly well suited for this purpose.
Notably, Kontsevich's formal period algebra represents a torsor
under the motivic Galois group in Nori's sense, and the period
conjecture of Kontsevich and Zagier can be recast in this setting.
Periods and Nori Motives is highly informative and will appeal to
graduate students interested in algebraic geometry and number
theory as well as researchers working in related fields. Containing
relevant background material on topics such as singular cohomology,
algebraic de Rham cohomology, diagram categories and rigid tensor
categories, as well as many interesting examples, the overall
presentation of this book is self-contained.
This exploration of the relation between periods and transcendental
numbers brings Baker's theory of linear forms in logarithms into
its most general framework, the theory of 1-motives. Written by
leading experts in the field, it contains original results and
finalises the theory of linear relations of 1-periods, answering
long-standing questions in transcendence theory. It provides a
complete exposition of the new theory for researchers, but also
serves as an introduction to transcendence for graduate students
and newcomers. It begins with foundational material, including a
review of the theory of commutative algebraic groups and the
analytic subgroup theorem as well as the basics of singular
homology and de Rham cohomology. Part II addresses periods of
1-motives, linking back to classical examples like the
transcendence of , before the authors turn to periods of algebraic
varieties in Part III. Finally, Part IV aims at a dimension formula
for the space of periods of a 1-motive in terms of its data.
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